What is (\partial)/(\partial y)((4x-y)/(4x+y)) ?

The answer to (\partial)/(\partial y)((4x-y)/(4x+y)) is -(8x)/((4x+y)^2)

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(\partial)/(\partial y)((4x-y)/(4x+y))

\frac{\partial}{\partial y} (\frac{4 x - y}{4 x + y})

- \frac{8 x}{( 4 x + y ) ^{2}}

Solution steps

\frac{\partial}{\partial y} (\frac{4 x - y}{4 x + y})

Treat

x

as a constant

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{\partial}{\partial y} ( 4 x - y ) ( 4 x + y ) - \frac{\partial}{\partial y} ( 4 x + y ) ( 4 x - y )}{( 4 x + y ) ^{2}}

\frac{\partial}{\partial y} (4 x - y) = - 1

\frac{\partial}{\partial y} (4 x + y) = 1

= \frac{( - 1 ) ( 4 x + y ) - 1 \cdot ( 4 x - y )}{( 4 x + y ) ^{2}}

Simplify

\frac{( - 1 ) ( 4 x + y ) - 1 \cdot ( 4 x - y )}{( 4 x + y ) ^{2}} : - \frac{8 x}{( 4 x + y ) ^{2}}

= - \frac{8 x}{( 4 x + y ) ^{2}}

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What is (\partial)/(\partial y)((4x-y)/(4x+y)) ?
The answer to (\partial)/(\partial y)((4x-y)/(4x+y)) is -(8x)/((4x+y)^2)